Identifier
Mp00014:
Binary trees
—to 132-avoiding permutation⟶
Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Images
=>
Cc0010;cc-rep-0Cc0010;cc-rep-3
[.,.]=>[1]=>[1]=>[.,.]
[.,[.,.]]=>[2,1]=>[2,1]=>[[.,.],.]
[[.,.],.]=>[1,2]=>[1,2]=>[.,[.,.]]
[.,[.,[.,.]]]=>[3,2,1]=>[2,3,1]=>[[.,[.,.]],.]
[.,[[.,.],.]]=>[2,3,1]=>[3,2,1]=>[[[.,.],.],.]
[[.,.],[.,.]]=>[3,1,2]=>[3,1,2]=>[[.,.],[.,.]]
[[.,[.,.]],.]=>[2,1,3]=>[2,1,3]=>[[.,.],[.,.]]
[[[.,.],.],.]=>[1,2,3]=>[1,2,3]=>[.,[.,[.,.]]]
[.,[.,[.,[.,.]]]]=>[4,3,2,1]=>[2,3,4,1]=>[[.,[.,[.,.]]],.]
[.,[.,[[.,.],.]]]=>[3,4,2,1]=>[2,4,3,1]=>[[.,[[.,.],.]],.]
[.,[[.,.],[.,.]]]=>[4,2,3,1]=>[3,4,2,1]=>[[[.,[.,.]],.],.]
[.,[[.,[.,.]],.]]=>[3,2,4,1]=>[4,3,2,1]=>[[[[.,.],.],.],.]
[.,[[[.,.],.],.]]=>[2,3,4,1]=>[4,2,3,1]=>[[[.,.],[.,.]],.]
[[.,.],[.,[.,.]]]=>[4,3,1,2]=>[3,1,4,2]=>[[.,.],[[.,.],.]]
[[.,.],[[.,.],.]]=>[3,4,1,2]=>[4,1,3,2]=>[[.,.],[[.,.],.]]
[[.,[.,.]],[.,.]]=>[4,2,1,3]=>[2,4,1,3]=>[[.,[.,.]],[.,.]]
[[[.,.],.],[.,.]]=>[4,1,2,3]=>[4,1,2,3]=>[[.,.],[.,[.,.]]]
[[.,[.,[.,.]]],.]=>[3,2,1,4]=>[2,3,1,4]=>[[.,[.,.]],[.,.]]
[[.,[[.,.],.]],.]=>[2,3,1,4]=>[3,2,1,4]=>[[[.,.],.],[.,.]]
[[[.,.],[.,.]],.]=>[3,1,2,4]=>[3,1,2,4]=>[[.,.],[.,[.,.]]]
[[[.,[.,.]],.],.]=>[2,1,3,4]=>[2,1,3,4]=>[[.,.],[.,[.,.]]]
[[[[.,.],.],.],.]=>[1,2,3,4]=>[1,2,3,4]=>[.,[.,[.,[.,.]]]]
[.,[.,[.,[.,[.,.]]]]]=>[5,4,3,2,1]=>[2,3,4,5,1]=>[[.,[.,[.,[.,.]]]],.]
[.,[.,[.,[[.,.],.]]]]=>[4,5,3,2,1]=>[2,3,5,4,1]=>[[.,[.,[[.,.],.]]],.]
[.,[.,[[.,.],[.,.]]]]=>[5,3,4,2,1]=>[2,4,5,3,1]=>[[.,[[.,[.,.]],.]],.]
[.,[.,[[.,[.,.]],.]]]=>[4,3,5,2,1]=>[2,5,4,3,1]=>[[.,[[[.,.],.],.]],.]
[.,[.,[[[.,.],.],.]]]=>[3,4,5,2,1]=>[2,5,3,4,1]=>[[.,[[.,.],[.,.]]],.]
[.,[[.,.],[.,[.,.]]]]=>[5,4,2,3,1]=>[3,4,2,5,1]=>[[[.,[.,.]],[.,.]],.]
[.,[[.,.],[[.,.],.]]]=>[4,5,2,3,1]=>[3,5,2,4,1]=>[[[.,[.,.]],[.,.]],.]
[.,[[.,[.,.]],[.,.]]]=>[5,3,2,4,1]=>[4,3,5,2,1]=>[[[[.,.],[.,.]],.],.]
[.,[[[.,.],.],[.,.]]]=>[5,2,3,4,1]=>[4,5,2,3,1]=>[[[.,[.,.]],[.,.]],.]
[.,[[.,[.,[.,.]]],.]]=>[4,3,2,5,1]=>[5,3,4,2,1]=>[[[[.,.],[.,.]],.],.]
[.,[[.,[[.,.],.]],.]]=>[3,4,2,5,1]=>[5,4,3,2,1]=>[[[[[.,.],.],.],.],.]
[.,[[[.,.],[.,.]],.]]=>[4,2,3,5,1]=>[5,4,2,3,1]=>[[[[.,.],.],[.,.]],.]
[.,[[[.,[.,.]],.],.]]=>[3,2,4,5,1]=>[5,3,2,4,1]=>[[[[.,.],.],[.,.]],.]
[.,[[[[.,.],.],.],.]]=>[2,3,4,5,1]=>[5,2,3,4,1]=>[[[.,.],[.,[.,.]]],.]
[[.,.],[.,[.,[.,.]]]]=>[5,4,3,1,2]=>[3,1,4,5,2]=>[[.,.],[[.,[.,.]],.]]
[[.,.],[.,[[.,.],.]]]=>[4,5,3,1,2]=>[3,1,5,4,2]=>[[.,.],[[[.,.],.],.]]
[[.,.],[[.,.],[.,.]]]=>[5,3,4,1,2]=>[4,1,5,3,2]=>[[.,.],[[[.,.],.],.]]
[[.,.],[[.,[.,.]],.]]=>[4,3,5,1,2]=>[5,1,4,3,2]=>[[.,.],[[[.,.],.],.]]
[[.,.],[[[.,.],.],.]]=>[3,4,5,1,2]=>[5,1,3,4,2]=>[[.,.],[[.,[.,.]],.]]
[[.,[.,.]],[.,[.,.]]]=>[5,4,2,1,3]=>[2,4,1,5,3]=>[[.,[.,.]],[[.,.],.]]
[[.,[.,.]],[[.,.],.]]=>[4,5,2,1,3]=>[2,5,1,4,3]=>[[.,[.,.]],[[.,.],.]]
[[[.,.],.],[.,[.,.]]]=>[5,4,1,2,3]=>[4,1,2,5,3]=>[[.,.],[.,[[.,.],.]]]
[[[.,.],.],[[.,.],.]]=>[4,5,1,2,3]=>[5,1,2,4,3]=>[[.,.],[.,[[.,.],.]]]
[[.,[.,[.,.]]],[.,.]]=>[5,3,2,1,4]=>[2,3,5,1,4]=>[[.,[.,[.,.]]],[.,.]]
[[.,[[.,.],.]],[.,.]]=>[5,2,3,1,4]=>[3,5,2,1,4]=>[[[.,[.,.]],.],[.,.]]
[[[.,.],[.,.]],[.,.]]=>[5,3,1,2,4]=>[3,1,5,2,4]=>[[.,.],[[.,.],[.,.]]]
[[[.,[.,.]],.],[.,.]]=>[5,2,1,3,4]=>[2,5,1,3,4]=>[[.,[.,.]],[.,[.,.]]]
[[[[.,.],.],.],[.,.]]=>[5,1,2,3,4]=>[5,1,2,3,4]=>[[.,.],[.,[.,[.,.]]]]
[[.,[.,[.,[.,.]]]],.]=>[4,3,2,1,5]=>[2,3,4,1,5]=>[[.,[.,[.,.]]],[.,.]]
[[.,[.,[[.,.],.]]],.]=>[3,4,2,1,5]=>[2,4,3,1,5]=>[[.,[[.,.],.]],[.,.]]
[[.,[[.,.],[.,.]]],.]=>[4,2,3,1,5]=>[3,4,2,1,5]=>[[[.,[.,.]],.],[.,.]]
[[.,[[.,[.,.]],.]],.]=>[3,2,4,1,5]=>[4,3,2,1,5]=>[[[[.,.],.],.],[.,.]]
[[.,[[[.,.],.],.]],.]=>[2,3,4,1,5]=>[4,2,3,1,5]=>[[[.,.],[.,.]],[.,.]]
[[[.,.],[.,[.,.]]],.]=>[4,3,1,2,5]=>[3,1,4,2,5]=>[[.,.],[[.,.],[.,.]]]
[[[.,.],[[.,.],.]],.]=>[3,4,1,2,5]=>[4,1,3,2,5]=>[[.,.],[[.,.],[.,.]]]
[[[.,[.,.]],[.,.]],.]=>[4,2,1,3,5]=>[2,4,1,3,5]=>[[.,[.,.]],[.,[.,.]]]
[[[[.,.],.],[.,.]],.]=>[4,1,2,3,5]=>[4,1,2,3,5]=>[[.,.],[.,[.,[.,.]]]]
[[[.,[.,[.,.]]],.],.]=>[3,2,1,4,5]=>[2,3,1,4,5]=>[[.,[.,.]],[.,[.,.]]]
[[[.,[[.,.],.]],.],.]=>[2,3,1,4,5]=>[3,2,1,4,5]=>[[[.,.],.],[.,[.,.]]]
[[[[.,.],[.,.]],.],.]=>[3,1,2,4,5]=>[3,1,2,4,5]=>[[.,.],[.,[.,[.,.]]]]
[[[[.,[.,.]],.],.],.]=>[2,1,3,4,5]=>[2,1,3,4,5]=>[[.,.],[.,[.,[.,.]]]]
[[[[[.,.],.],.],.],.]=>[1,2,3,4,5]=>[1,2,3,4,5]=>[.,[.,[.,[.,[.,.]]]]]
[.,[.,[.,[.,[.,[.,.]]]]]]=>[6,5,4,3,2,1]=>[2,3,4,5,6,1]=>[[.,[.,[.,[.,[.,.]]]]],.]
[.,[.,[.,[.,[[.,.],.]]]]]=>[5,6,4,3,2,1]=>[2,3,4,6,5,1]=>[[.,[.,[.,[[.,.],.]]]],.]
[.,[.,[.,[[.,.],[.,.]]]]]=>[6,4,5,3,2,1]=>[2,3,5,6,4,1]=>[[.,[.,[[.,[.,.]],.]]],.]
[.,[.,[.,[[.,[.,.]],.]]]]=>[5,4,6,3,2,1]=>[2,3,6,5,4,1]=>[[.,[.,[[[.,.],.],.]]],.]
[.,[.,[.,[[[.,.],.],.]]]]=>[4,5,6,3,2,1]=>[2,3,6,4,5,1]=>[[.,[.,[[.,.],[.,.]]]],.]
[.,[.,[[.,.],[.,[.,.]]]]]=>[6,5,3,4,2,1]=>[2,4,5,3,6,1]=>[[.,[[.,[.,.]],[.,.]]],.]
[.,[.,[[.,.],[[.,.],.]]]]=>[5,6,3,4,2,1]=>[2,4,6,3,5,1]=>[[.,[[.,[.,.]],[.,.]]],.]
[.,[.,[[.,[.,.]],[.,.]]]]=>[6,4,3,5,2,1]=>[2,5,4,6,3,1]=>[[.,[[[.,.],[.,.]],.]],.]
[.,[.,[[[.,.],.],[.,.]]]]=>[6,3,4,5,2,1]=>[2,5,6,3,4,1]=>[[.,[[.,[.,.]],[.,.]]],.]
[.,[.,[[.,[.,[.,.]]],.]]]=>[5,4,3,6,2,1]=>[2,6,4,5,3,1]=>[[.,[[[.,.],[.,.]],.]],.]
[.,[.,[[.,[[.,.],.]],.]]]=>[4,5,3,6,2,1]=>[2,6,5,4,3,1]=>[[.,[[[[.,.],.],.],.]],.]
[.,[.,[[[.,.],[.,.]],.]]]=>[5,3,4,6,2,1]=>[2,6,5,3,4,1]=>[[.,[[[.,.],.],[.,.]]],.]
[.,[.,[[[.,[.,.]],.],.]]]=>[4,3,5,6,2,1]=>[2,6,4,3,5,1]=>[[.,[[[.,.],.],[.,.]]],.]
[.,[.,[[[[.,.],.],.],.]]]=>[3,4,5,6,2,1]=>[2,6,3,4,5,1]=>[[.,[[.,.],[.,[.,.]]]],.]
[.,[[.,.],[.,[.,[.,.]]]]]=>[6,5,4,2,3,1]=>[3,4,2,5,6,1]=>[[[.,[.,.]],[.,[.,.]]],.]
[.,[[.,.],[.,[[.,.],.]]]]=>[5,6,4,2,3,1]=>[3,4,2,6,5,1]=>[[[.,[.,.]],[[.,.],.]],.]
[.,[[.,.],[[.,.],[.,.]]]]=>[6,4,5,2,3,1]=>[3,5,2,6,4,1]=>[[[.,[.,.]],[[.,.],.]],.]
[.,[[.,.],[[.,[.,.]],.]]]=>[5,4,6,2,3,1]=>[3,6,2,5,4,1]=>[[[.,[.,.]],[[.,.],.]],.]
[.,[[.,.],[[[.,.],.],.]]]=>[4,5,6,2,3,1]=>[3,6,2,4,5,1]=>[[[.,[.,.]],[.,[.,.]]],.]
[.,[[.,[.,.]],[.,[.,.]]]]=>[6,5,3,2,4,1]=>[4,3,5,2,6,1]=>[[[[.,.],[.,.]],[.,.]],.]
[.,[[.,[.,.]],[[.,.],.]]]=>[5,6,3,2,4,1]=>[4,3,6,2,5,1]=>[[[[.,.],[.,.]],[.,.]],.]
[.,[[[.,.],.],[.,[.,.]]]]=>[6,5,2,3,4,1]=>[4,5,2,3,6,1]=>[[[.,[.,.]],[.,[.,.]]],.]
[.,[[[.,.],.],[[.,.],.]]]=>[5,6,2,3,4,1]=>[4,6,2,3,5,1]=>[[[.,[.,.]],[.,[.,.]]],.]
[.,[[.,[.,[.,.]]],[.,.]]]=>[6,4,3,2,5,1]=>[5,3,4,6,2,1]=>[[[[.,.],[.,[.,.]]],.],.]
[.,[[.,[[.,.],.]],[.,.]]]=>[6,3,4,2,5,1]=>[5,4,6,3,2,1]=>[[[[[.,.],[.,.]],.],.],.]
[.,[[[.,.],[.,.]],[.,.]]]=>[6,4,2,3,5,1]=>[5,4,2,6,3,1]=>[[[[.,.],.],[[.,.],.]],.]
[.,[[[.,[.,.]],.],[.,.]]]=>[6,3,2,4,5,1]=>[5,3,6,2,4,1]=>[[[[.,.],[.,.]],[.,.]],.]
[.,[[[[.,.],.],.],[.,.]]]=>[6,2,3,4,5,1]=>[5,6,2,3,4,1]=>[[[.,[.,.]],[.,[.,.]]],.]
[.,[[.,[.,[.,[.,.]]]],.]]=>[5,4,3,2,6,1]=>[6,3,4,5,2,1]=>[[[[.,.],[.,[.,.]]],.],.]
[.,[[.,[.,[[.,.],.]]],.]]=>[4,5,3,2,6,1]=>[6,3,5,4,2,1]=>[[[[.,.],[[.,.],.]],.],.]
[.,[[.,[[.,.],[.,.]]],.]]=>[5,3,4,2,6,1]=>[6,4,5,3,2,1]=>[[[[[.,.],[.,.]],.],.],.]
[.,[[.,[[.,[.,.]],.]],.]]=>[4,3,5,2,6,1]=>[6,5,4,3,2,1]=>[[[[[[.,.],.],.],.],.],.]
[.,[[.,[[[.,.],.],.]],.]]=>[3,4,5,2,6,1]=>[6,5,3,4,2,1]=>[[[[[.,.],.],[.,.]],.],.]
[.,[[[.,.],[.,[.,.]]],.]]=>[5,4,2,3,6,1]=>[6,4,2,5,3,1]=>[[[[.,.],.],[[.,.],.]],.]
[.,[[[.,.],[[.,.],.]],.]]=>[4,5,2,3,6,1]=>[6,5,2,4,3,1]=>[[[[.,.],.],[[.,.],.]],.]
[.,[[[.,[.,.]],[.,.]],.]]=>[5,3,2,4,6,1]=>[6,3,5,2,4,1]=>[[[[.,.],[.,.]],[.,.]],.]
[.,[[[[.,.],.],[.,.]],.]]=>[5,2,3,4,6,1]=>[6,5,2,3,4,1]=>[[[[.,.],.],[.,[.,.]]],.]
[.,[[[.,[.,[.,.]]],.],.]]=>[4,3,2,5,6,1]=>[6,3,4,2,5,1]=>[[[[.,.],[.,.]],[.,.]],.]
[.,[[[.,[[.,.],.]],.],.]]=>[3,4,2,5,6,1]=>[6,4,3,2,5,1]=>[[[[[.,.],.],.],[.,.]],.]
[.,[[[[.,.],[.,.]],.],.]]=>[4,2,3,5,6,1]=>[6,4,2,3,5,1]=>[[[[.,.],.],[.,[.,.]]],.]
[.,[[[[.,[.,.]],.],.],.]]=>[3,2,4,5,6,1]=>[6,3,2,4,5,1]=>[[[[.,.],.],[.,[.,.]]],.]
[.,[[[[[.,.],.],.],.],.]]=>[2,3,4,5,6,1]=>[6,2,3,4,5,1]=>[[[.,.],[.,[.,[.,.]]]],.]
[[.,.],[.,[.,[.,[.,.]]]]]=>[6,5,4,3,1,2]=>[3,1,4,5,6,2]=>[[.,.],[[.,[.,[.,.]]],.]]
[[.,.],[.,[.,[[.,.],.]]]]=>[5,6,4,3,1,2]=>[3,1,4,6,5,2]=>[[.,.],[[.,[[.,.],.]],.]]
[[.,.],[.,[[.,.],[.,.]]]]=>[6,4,5,3,1,2]=>[3,1,5,6,4,2]=>[[.,.],[[[.,[.,.]],.],.]]
[[.,.],[.,[[.,[.,.]],.]]]=>[5,4,6,3,1,2]=>[3,1,6,5,4,2]=>[[.,.],[[[[.,.],.],.],.]]
[[.,.],[.,[[[.,.],.],.]]]=>[4,5,6,3,1,2]=>[3,1,6,4,5,2]=>[[.,.],[[[.,.],[.,.]],.]]
[[.,.],[[.,.],[.,[.,.]]]]=>[6,5,3,4,1,2]=>[4,1,5,3,6,2]=>[[.,.],[[[.,.],[.,.]],.]]
[[.,.],[[.,.],[[.,.],.]]]=>[5,6,3,4,1,2]=>[4,1,6,3,5,2]=>[[.,.],[[[.,.],[.,.]],.]]
[[.,.],[[.,[.,.]],[.,.]]]=>[6,4,3,5,1,2]=>[5,1,4,6,3,2]=>[[.,.],[[[.,[.,.]],.],.]]
[[.,.],[[[.,.],.],[.,.]]]=>[6,3,4,5,1,2]=>[5,1,6,3,4,2]=>[[.,.],[[[.,.],[.,.]],.]]
[[.,.],[[.,[.,[.,.]]],.]]=>[5,4,3,6,1,2]=>[6,1,4,5,3,2]=>[[.,.],[[[.,[.,.]],.],.]]
[[.,.],[[.,[[.,.],.]],.]]=>[4,5,3,6,1,2]=>[6,1,5,4,3,2]=>[[.,.],[[[[.,.],.],.],.]]
[[.,.],[[[.,.],[.,.]],.]]=>[5,3,4,6,1,2]=>[6,1,5,3,4,2]=>[[.,.],[[[.,.],[.,.]],.]]
[[.,.],[[[.,[.,.]],.],.]]=>[4,3,5,6,1,2]=>[6,1,4,3,5,2]=>[[.,.],[[[.,.],[.,.]],.]]
[[.,.],[[[[.,.],.],.],.]]=>[3,4,5,6,1,2]=>[6,1,3,4,5,2]=>[[.,.],[[.,[.,[.,.]]],.]]
[[.,[.,.]],[.,[.,[.,.]]]]=>[6,5,4,2,1,3]=>[2,4,1,5,6,3]=>[[.,[.,.]],[[.,[.,.]],.]]
[[.,[.,.]],[.,[[.,.],.]]]=>[5,6,4,2,1,3]=>[2,4,1,6,5,3]=>[[.,[.,.]],[[[.,.],.],.]]
[[.,[.,.]],[[.,.],[.,.]]]=>[6,4,5,2,1,3]=>[2,5,1,6,4,3]=>[[.,[.,.]],[[[.,.],.],.]]
[[.,[.,.]],[[.,[.,.]],.]]=>[5,4,6,2,1,3]=>[2,6,1,5,4,3]=>[[.,[.,.]],[[[.,.],.],.]]
[[.,[.,.]],[[[.,.],.],.]]=>[4,5,6,2,1,3]=>[2,6,1,4,5,3]=>[[.,[.,.]],[[.,[.,.]],.]]
[[[.,.],.],[.,[.,[.,.]]]]=>[6,5,4,1,2,3]=>[4,1,2,5,6,3]=>[[.,.],[.,[[.,[.,.]],.]]]
[[[.,.],.],[.,[[.,.],.]]]=>[5,6,4,1,2,3]=>[4,1,2,6,5,3]=>[[.,.],[.,[[[.,.],.],.]]]
[[[.,.],.],[[.,.],[.,.]]]=>[6,4,5,1,2,3]=>[5,1,2,6,4,3]=>[[.,.],[.,[[[.,.],.],.]]]
[[[.,.],.],[[.,[.,.]],.]]=>[5,4,6,1,2,3]=>[6,1,2,5,4,3]=>[[.,.],[.,[[[.,.],.],.]]]
[[[.,.],.],[[[.,.],.],.]]=>[4,5,6,1,2,3]=>[6,1,2,4,5,3]=>[[.,.],[.,[[.,[.,.]],.]]]
[[.,[.,[.,.]]],[.,[.,.]]]=>[6,5,3,2,1,4]=>[2,3,5,1,6,4]=>[[.,[.,[.,.]]],[[.,.],.]]
[[.,[.,[.,.]]],[[.,.],.]]=>[5,6,3,2,1,4]=>[2,3,6,1,5,4]=>[[.,[.,[.,.]]],[[.,.],.]]
[[.,[[.,.],.]],[.,[.,.]]]=>[6,5,2,3,1,4]=>[3,5,2,1,6,4]=>[[[.,[.,.]],.],[[.,.],.]]
[[.,[[.,.],.]],[[.,.],.]]=>[5,6,2,3,1,4]=>[3,6,2,1,5,4]=>[[[.,[.,.]],.],[[.,.],.]]
[[[.,.],[.,.]],[.,[.,.]]]=>[6,5,3,1,2,4]=>[3,1,5,2,6,4]=>[[.,.],[[.,.],[[.,.],.]]]
[[[.,.],[.,.]],[[.,.],.]]=>[5,6,3,1,2,4]=>[3,1,6,2,5,4]=>[[.,.],[[.,.],[[.,.],.]]]
[[[.,[.,.]],.],[.,[.,.]]]=>[6,5,2,1,3,4]=>[2,5,1,3,6,4]=>[[.,[.,.]],[.,[[.,.],.]]]
[[[.,[.,.]],.],[[.,.],.]]=>[5,6,2,1,3,4]=>[2,6,1,3,5,4]=>[[.,[.,.]],[.,[[.,.],.]]]
[[[[.,.],.],.],[.,[.,.]]]=>[6,5,1,2,3,4]=>[5,1,2,3,6,4]=>[[.,.],[.,[.,[[.,.],.]]]]
[[[[.,.],.],.],[[.,.],.]]=>[5,6,1,2,3,4]=>[6,1,2,3,5,4]=>[[.,.],[.,[.,[[.,.],.]]]]
[[.,[.,[.,[.,.]]]],[.,.]]=>[6,4,3,2,1,5]=>[2,3,4,6,1,5]=>[[.,[.,[.,[.,.]]]],[.,.]]
[[.,[.,[[.,.],.]]],[.,.]]=>[6,3,4,2,1,5]=>[2,4,6,3,1,5]=>[[.,[[.,[.,.]],.]],[.,.]]
[[.,[[.,.],[.,.]]],[.,.]]=>[6,4,2,3,1,5]=>[3,4,2,6,1,5]=>[[[.,[.,.]],[.,.]],[.,.]]
[[.,[[.,[.,.]],.]],[.,.]]=>[6,3,2,4,1,5]=>[4,3,6,2,1,5]=>[[[[.,.],[.,.]],.],[.,.]]
[[.,[[[.,.],.],.]],[.,.]]=>[6,2,3,4,1,5]=>[4,6,2,3,1,5]=>[[[.,[.,.]],[.,.]],[.,.]]
[[[.,.],[.,[.,.]]],[.,.]]=>[6,4,3,1,2,5]=>[3,1,4,6,2,5]=>[[.,.],[[.,[.,.]],[.,.]]]
[[[.,.],[[.,.],.]],[.,.]]=>[6,3,4,1,2,5]=>[4,1,6,3,2,5]=>[[.,.],[[[.,.],.],[.,.]]]
[[[.,[.,.]],[.,.]],[.,.]]=>[6,4,2,1,3,5]=>[2,4,1,6,3,5]=>[[.,[.,.]],[[.,.],[.,.]]]
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[[[.,.],.],[[.,.],[[.,.],.]]]=>[6,7,4,5,1,2,3]=>[5,1,2,7,4,6,3]=>[[.,.],[.,[[[.,.],[.,.]],.]]]
[[[.,.],.],[[.,[.,.]],[.,.]]]=>[7,5,4,6,1,2,3]=>[6,1,2,5,7,4,3]=>[[.,.],[.,[[[.,[.,.]],.],.]]]
[[[.,.],.],[[[.,.],.],[.,.]]]=>[7,4,5,6,1,2,3]=>[6,1,2,7,4,5,3]=>[[.,.],[.,[[[.,.],[.,.]],.]]]
[[[.,.],.],[[.,[.,[.,.]]],.]]=>[6,5,4,7,1,2,3]=>[7,1,2,5,6,4,3]=>[[.,.],[.,[[[.,[.,.]],.],.]]]
[[[.,.],.],[[.,[[.,.],.]],.]]=>[5,6,4,7,1,2,3]=>[7,1,2,6,5,4,3]=>[[.,.],[.,[[[[.,.],.],.],.]]]
[[[.,.],.],[[[.,.],[.,.]],.]]=>[6,4,5,7,1,2,3]=>[7,1,2,6,4,5,3]=>[[.,.],[.,[[[.,.],[.,.]],.]]]
[[[.,.],.],[[[.,[.,.]],.],.]]=>[5,4,6,7,1,2,3]=>[7,1,2,5,4,6,3]=>[[.,.],[.,[[[.,.],[.,.]],.]]]
[[[.,.],.],[[[[.,.],.],.],.]]=>[4,5,6,7,1,2,3]=>[7,1,2,4,5,6,3]=>[[.,.],[.,[[.,[.,[.,.]]],.]]]
[[.,[.,[.,.]]],[.,[.,[.,.]]]]=>[7,6,5,3,2,1,4]=>[2,3,5,1,6,7,4]=>[[.,[.,[.,.]]],[[.,[.,.]],.]]
[[.,[.,[.,.]]],[.,[[.,.],.]]]=>[6,7,5,3,2,1,4]=>[2,3,5,1,7,6,4]=>[[.,[.,[.,.]]],[[[.,.],.],.]]
[[.,[.,[.,.]]],[[.,.],[.,.]]]=>[7,5,6,3,2,1,4]=>[2,3,6,1,7,5,4]=>[[.,[.,[.,.]]],[[[.,.],.],.]]
[[.,[.,[.,.]]],[[.,[.,.]],.]]=>[6,5,7,3,2,1,4]=>[2,3,7,1,6,5,4]=>[[.,[.,[.,.]]],[[[.,.],.],.]]
[[.,[.,[.,.]]],[[[.,.],.],.]]=>[5,6,7,3,2,1,4]=>[2,3,7,1,5,6,4]=>[[.,[.,[.,.]]],[[.,[.,.]],.]]
[[.,[[.,.],.]],[.,[.,[.,.]]]]=>[7,6,5,2,3,1,4]=>[3,5,2,1,6,7,4]=>[[[.,[.,.]],.],[[.,[.,.]],.]]
[[.,[[.,.],.]],[.,[[.,.],.]]]=>[6,7,5,2,3,1,4]=>[3,5,2,1,7,6,4]=>[[[.,[.,.]],.],[[[.,.],.],.]]
[[.,[[.,.],.]],[[.,.],[.,.]]]=>[7,5,6,2,3,1,4]=>[3,6,2,1,7,5,4]=>[[[.,[.,.]],.],[[[.,.],.],.]]
[[.,[[.,.],.]],[[.,[.,.]],.]]=>[6,5,7,2,3,1,4]=>[3,7,2,1,6,5,4]=>[[[.,[.,.]],.],[[[.,.],.],.]]
[[.,[[.,.],.]],[[[.,.],.],.]]=>[5,6,7,2,3,1,4]=>[3,7,2,1,5,6,4]=>[[[.,[.,.]],.],[[.,[.,.]],.]]
[[[.,.],[.,.]],[.,[.,[.,.]]]]=>[7,6,5,3,1,2,4]=>[3,1,5,2,6,7,4]=>[[.,.],[[.,.],[[.,[.,.]],.]]]
[[[.,.],[.,.]],[.,[[.,.],.]]]=>[6,7,5,3,1,2,4]=>[3,1,5,2,7,6,4]=>[[.,.],[[.,.],[[[.,.],.],.]]]
[[[.,.],[.,.]],[[.,.],[.,.]]]=>[7,5,6,3,1,2,4]=>[3,1,6,2,7,5,4]=>[[.,.],[[.,.],[[[.,.],.],.]]]
[[[.,.],[.,.]],[[.,[.,.]],.]]=>[6,5,7,3,1,2,4]=>[3,1,7,2,6,5,4]=>[[.,.],[[.,.],[[[.,.],.],.]]]
[[[.,.],[.,.]],[[[.,.],.],.]]=>[5,6,7,3,1,2,4]=>[3,1,7,2,5,6,4]=>[[.,.],[[.,.],[[.,[.,.]],.]]]
[[[.,[.,.]],.],[.,[.,[.,.]]]]=>[7,6,5,2,1,3,4]=>[2,5,1,3,6,7,4]=>[[.,[.,.]],[.,[[.,[.,.]],.]]]
[[[.,[.,.]],.],[.,[[.,.],.]]]=>[6,7,5,2,1,3,4]=>[2,5,1,3,7,6,4]=>[[.,[.,.]],[.,[[[.,.],.],.]]]
[[[.,[.,.]],.],[[.,.],[.,.]]]=>[7,5,6,2,1,3,4]=>[2,6,1,3,7,5,4]=>[[.,[.,.]],[.,[[[.,.],.],.]]]
[[[.,[.,.]],.],[[.,[.,.]],.]]=>[6,5,7,2,1,3,4]=>[2,7,1,3,6,5,4]=>[[.,[.,.]],[.,[[[.,.],.],.]]]
[[[.,[.,.]],.],[[[.,.],.],.]]=>[5,6,7,2,1,3,4]=>[2,7,1,3,5,6,4]=>[[.,[.,.]],[.,[[.,[.,.]],.]]]
[[[[.,.],.],.],[.,[.,[.,.]]]]=>[7,6,5,1,2,3,4]=>[5,1,2,3,6,7,4]=>[[.,.],[.,[.,[[.,[.,.]],.]]]]
[[[[.,.],.],.],[.,[[.,.],.]]]=>[6,7,5,1,2,3,4]=>[5,1,2,3,7,6,4]=>[[.,.],[.,[.,[[[.,.],.],.]]]]
[[[[.,.],.],.],[[.,.],[.,.]]]=>[7,5,6,1,2,3,4]=>[6,1,2,3,7,5,4]=>[[.,.],[.,[.,[[[.,.],.],.]]]]
[[[[.,.],.],.],[[.,[.,.]],.]]=>[6,5,7,1,2,3,4]=>[7,1,2,3,6,5,4]=>[[.,.],[.,[.,[[[.,.],.],.]]]]
[[[[.,.],.],.],[[[.,.],.],.]]=>[5,6,7,1,2,3,4]=>[7,1,2,3,5,6,4]=>[[.,.],[.,[.,[[.,[.,.]],.]]]]
[[.,[.,[.,[.,.]]]],[.,[.,.]]]=>[7,6,4,3,2,1,5]=>[2,3,4,6,1,7,5]=>[[.,[.,[.,[.,.]]]],[[.,.],.]]
[[.,[.,[.,[.,.]]]],[[.,.],.]]=>[6,7,4,3,2,1,5]=>[2,3,4,7,1,6,5]=>[[.,[.,[.,[.,.]]]],[[.,.],.]]
[[.,[.,[[.,.],.]]],[.,[.,.]]]=>[7,6,3,4,2,1,5]=>[2,4,6,3,1,7,5]=>[[.,[[.,[.,.]],.]],[[.,.],.]]
[[.,[.,[[.,.],.]]],[[.,.],.]]=>[6,7,3,4,2,1,5]=>[2,4,7,3,1,6,5]=>[[.,[[.,[.,.]],.]],[[.,.],.]]
[[.,[[.,.],[.,.]]],[.,[.,.]]]=>[7,6,4,2,3,1,5]=>[3,4,2,6,1,7,5]=>[[[.,[.,.]],[.,.]],[[.,.],.]]
[[.,[[.,.],[.,.]]],[[.,.],.]]=>[6,7,4,2,3,1,5]=>[3,4,2,7,1,6,5]=>[[[.,[.,.]],[.,.]],[[.,.],.]]
[[.,[[.,[.,.]],.]],[.,[.,.]]]=>[7,6,3,2,4,1,5]=>[4,3,6,2,1,7,5]=>[[[[.,.],[.,.]],.],[[.,.],.]]
[[.,[[.,[.,.]],.]],[[.,.],.]]=>[6,7,3,2,4,1,5]=>[4,3,7,2,1,6,5]=>[[[[.,.],[.,.]],.],[[.,.],.]]
[[.,[[[.,.],.],.]],[.,[.,.]]]=>[7,6,2,3,4,1,5]=>[4,6,2,3,1,7,5]=>[[[.,[.,.]],[.,.]],[[.,.],.]]
[[.,[[[.,.],.],.]],[[.,.],.]]=>[6,7,2,3,4,1,5]=>[4,7,2,3,1,6,5]=>[[[.,[.,.]],[.,.]],[[.,.],.]]
[[[.,.],[.,[.,.]]],[.,[.,.]]]=>[7,6,4,3,1,2,5]=>[3,1,4,6,2,7,5]=>[[.,.],[[.,[.,.]],[[.,.],.]]]
[[[.,.],[.,[.,.]]],[[.,.],.]]=>[6,7,4,3,1,2,5]=>[3,1,4,7,2,6,5]=>[[.,.],[[.,[.,.]],[[.,.],.]]]
[[[.,.],[[.,.],.]],[.,[.,.]]]=>[7,6,3,4,1,2,5]=>[4,1,6,3,2,7,5]=>[[.,.],[[[.,.],.],[[.,.],.]]]
[[[.,.],[[.,.],.]],[[.,.],.]]=>[6,7,3,4,1,2,5]=>[4,1,7,3,2,6,5]=>[[.,.],[[[.,.],.],[[.,.],.]]]
[[[.,[.,.]],[.,.]],[.,[.,.]]]=>[7,6,4,2,1,3,5]=>[2,4,1,6,3,7,5]=>[[.,[.,.]],[[.,.],[[.,.],.]]]
[[[.,[.,.]],[.,.]],[[.,.],.]]=>[6,7,4,2,1,3,5]=>[2,4,1,7,3,6,5]=>[[.,[.,.]],[[.,.],[[.,.],.]]]
[[[[.,.],.],[.,.]],[.,[.,.]]]=>[7,6,4,1,2,3,5]=>[4,1,2,6,3,7,5]=>[[.,.],[.,[[.,.],[[.,.],.]]]]
[[[[.,.],.],[.,.]],[[.,.],.]]=>[6,7,4,1,2,3,5]=>[4,1,2,7,3,6,5]=>[[.,.],[.,[[.,.],[[.,.],.]]]]
[[[.,[.,[.,.]]],.],[.,[.,.]]]=>[7,6,3,2,1,4,5]=>[2,3,6,1,4,7,5]=>[[.,[.,[.,.]]],[.,[[.,.],.]]]
[[[.,[.,[.,.]]],.],[[.,.],.]]=>[6,7,3,2,1,4,5]=>[2,3,7,1,4,6,5]=>[[.,[.,[.,.]]],[.,[[.,.],.]]]
[[[.,[[.,.],.]],.],[.,[.,.]]]=>[7,6,2,3,1,4,5]=>[3,6,2,1,4,7,5]=>[[[.,[.,.]],.],[.,[[.,.],.]]]
[[[.,[[.,.],.]],.],[[.,.],.]]=>[6,7,2,3,1,4,5]=>[3,7,2,1,4,6,5]=>[[[.,[.,.]],.],[.,[[.,.],.]]]
[[[[.,.],[.,.]],.],[.,[.,.]]]=>[7,6,3,1,2,4,5]=>[3,1,6,2,4,7,5]=>[[.,.],[[.,.],[.,[[.,.],.]]]]
[[[[.,.],[.,.]],.],[[.,.],.]]=>[6,7,3,1,2,4,5]=>[3,1,7,2,4,6,5]=>[[.,.],[[.,.],[.,[[.,.],.]]]]
[[[[.,[.,.]],.],.],[.,[.,.]]]=>[7,6,2,1,3,4,5]=>[2,6,1,3,4,7,5]=>[[.,[.,.]],[.,[.,[[.,.],.]]]]
[[[[.,[.,.]],.],.],[[.,.],.]]=>[6,7,2,1,3,4,5]=>[2,7,1,3,4,6,5]=>[[.,[.,.]],[.,[.,[[.,.],.]]]]
[[[[[.,.],.],.],.],[.,[.,.]]]=>[7,6,1,2,3,4,5]=>[6,1,2,3,4,7,5]=>[[.,.],[.,[.,[.,[[.,.],.]]]]]
[[[[[.,.],.],.],.],[[.,.],.]]=>[6,7,1,2,3,4,5]=>[7,1,2,3,4,6,5]=>[[.,.],[.,[.,[.,[[.,.],.]]]]]
[[.,[.,[.,[.,[.,.]]]]],[.,.]]=>[7,5,4,3,2,1,6]=>[2,3,4,5,7,1,6]=>[[.,[.,[.,[.,[.,.]]]]],[.,.]]
[[.,[.,[.,[[.,.],.]]]],[.,.]]=>[7,4,5,3,2,1,6]=>[2,3,5,7,4,1,6]=>[[.,[.,[[.,[.,.]],.]]],[.,.]]
[[.,[.,[[.,.],[.,.]]]],[.,.]]=>[7,5,3,4,2,1,6]=>[2,4,5,3,7,1,6]=>[[.,[[.,[.,.]],[.,.]]],[.,.]]
[[.,[.,[[.,[.,.]],.]]],[.,.]]=>[7,4,3,5,2,1,6]=>[2,5,4,7,3,1,6]=>[[.,[[[.,.],[.,.]],.]],[.,.]]
[[.,[.,[[[.,.],.],.]]],[.,.]]=>[7,3,4,5,2,1,6]=>[2,5,7,3,4,1,6]=>[[.,[[.,[.,.]],[.,.]]],[.,.]]
[[.,[[.,.],[.,[.,.]]]],[.,.]]=>[7,5,4,2,3,1,6]=>[3,4,2,5,7,1,6]=>[[[.,[.,.]],[.,[.,.]]],[.,.]]
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[[.,[[.,[.,.]],[.,.]]],[.,.]]=>[7,5,3,2,4,1,6]=>[4,3,5,2,7,1,6]=>[[[[.,.],[.,.]],[.,.]],[.,.]]
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[[.,[[.,[.,[.,.]]],.]],[.,.]]=>[7,4,3,2,5,1,6]=>[5,3,4,7,2,1,6]=>[[[[.,.],[.,[.,.]]],.],[.,.]]
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[[.,[[[.,.],[.,.]],.]],[.,.]]=>[7,4,2,3,5,1,6]=>[5,4,2,7,3,1,6]=>[[[[.,.],.],[[.,.],.]],[.,.]]
[[.,[[[.,[.,.]],.],.]],[.,.]]=>[7,3,2,4,5,1,6]=>[5,3,7,2,4,1,6]=>[[[[.,.],[.,.]],[.,.]],[.,.]]
[[.,[[[[.,.],.],.],.]],[.,.]]=>[7,2,3,4,5,1,6]=>[5,7,2,3,4,1,6]=>[[[.,[.,.]],[.,[.,.]]],[.,.]]
[[[.,.],[.,[.,[.,.]]]],[.,.]]=>[7,5,4,3,1,2,6]=>[3,1,4,5,7,2,6]=>[[.,.],[[.,[.,[.,.]]],[.,.]]]
[[[.,.],[.,[[.,.],.]]],[.,.]]=>[7,4,5,3,1,2,6]=>[3,1,5,7,4,2,6]=>[[.,.],[[[.,[.,.]],.],[.,.]]]
[[[.,.],[[.,.],[.,.]]],[.,.]]=>[7,5,3,4,1,2,6]=>[4,1,5,3,7,2,6]=>[[.,.],[[[.,.],[.,.]],[.,.]]]
[[[.,.],[[.,[.,.]],.]],[.,.]]=>[7,4,3,5,1,2,6]=>[5,1,4,7,3,2,6]=>[[.,.],[[[.,[.,.]],.],[.,.]]]
[[[.,.],[[[.,.],.],.]],[.,.]]=>[7,3,4,5,1,2,6]=>[5,1,7,3,4,2,6]=>[[.,.],[[[.,.],[.,.]],[.,.]]]
[[[.,[.,.]],[.,[.,.]]],[.,.]]=>[7,5,4,2,1,3,6]=>[2,4,1,5,7,3,6]=>[[.,[.,.]],[[.,[.,.]],[.,.]]]
[[[.,[.,.]],[[.,.],.]],[.,.]]=>[7,4,5,2,1,3,6]=>[2,5,1,7,4,3,6]=>[[.,[.,.]],[[[.,.],.],[.,.]]]
[[[[.,.],.],[.,[.,.]]],[.,.]]=>[7,5,4,1,2,3,6]=>[4,1,2,5,7,3,6]=>[[.,.],[.,[[.,[.,.]],[.,.]]]]
[[[[.,.],.],[[.,.],.]],[.,.]]=>[7,4,5,1,2,3,6]=>[5,1,2,7,4,3,6]=>[[.,.],[.,[[[.,.],.],[.,.]]]]
[[[.,[.,[.,.]]],[.,.]],[.,.]]=>[7,5,3,2,1,4,6]=>[2,3,5,1,7,4,6]=>[[.,[.,[.,.]]],[[.,.],[.,.]]]
[[[.,[[.,.],.]],[.,.]],[.,.]]=>[7,5,2,3,1,4,6]=>[3,5,2,1,7,4,6]=>[[[.,[.,.]],.],[[.,.],[.,.]]]
[[[[.,.],[.,.]],[.,.]],[.,.]]=>[7,5,3,1,2,4,6]=>[3,1,5,2,7,4,6]=>[[.,.],[[.,.],[[.,.],[.,.]]]]
[[[[.,[.,.]],.],[.,.]],[.,.]]=>[7,5,2,1,3,4,6]=>[2,5,1,3,7,4,6]=>[[.,[.,.]],[.,[[.,.],[.,.]]]]
[[[[[.,.],.],.],[.,.]],[.,.]]=>[7,5,1,2,3,4,6]=>[5,1,2,3,7,4,6]=>[[.,.],[.,[.,[[.,.],[.,.]]]]]
[[[.,[.,[.,[.,.]]]],.],[.,.]]=>[7,4,3,2,1,5,6]=>[2,3,4,7,1,5,6]=>[[.,[.,[.,[.,.]]]],[.,[.,.]]]
[[[.,[.,[[.,.],.]]],.],[.,.]]=>[7,3,4,2,1,5,6]=>[2,4,7,3,1,5,6]=>[[.,[[.,[.,.]],.]],[.,[.,.]]]
[[[.,[[.,.],[.,.]]],.],[.,.]]=>[7,4,2,3,1,5,6]=>[3,4,2,7,1,5,6]=>[[[.,[.,.]],[.,.]],[.,[.,.]]]
[[[.,[[.,[.,.]],.]],.],[.,.]]=>[7,3,2,4,1,5,6]=>[4,3,7,2,1,5,6]=>[[[[.,.],[.,.]],.],[.,[.,.]]]
[[[.,[[[.,.],.],.]],.],[.,.]]=>[7,2,3,4,1,5,6]=>[4,7,2,3,1,5,6]=>[[[.,[.,.]],[.,.]],[.,[.,.]]]
[[[[.,.],[.,[.,.]]],.],[.,.]]=>[7,4,3,1,2,5,6]=>[3,1,4,7,2,5,6]=>[[.,.],[[.,[.,.]],[.,[.,.]]]]
[[[[.,.],[[.,.],.]],.],[.,.]]=>[7,3,4,1,2,5,6]=>[4,1,7,3,2,5,6]=>[[.,.],[[[.,.],.],[.,[.,.]]]]
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[[[[[.,.],.],[.,.]],.],[.,.]]=>[7,4,1,2,3,5,6]=>[4,1,2,7,3,5,6]=>[[.,.],[.,[[.,.],[.,[.,.]]]]]
[[[[.,[.,[.,.]]],.],.],[.,.]]=>[7,3,2,1,4,5,6]=>[2,3,7,1,4,5,6]=>[[.,[.,[.,.]]],[.,[.,[.,.]]]]
[[[[.,[[.,.],.]],.],.],[.,.]]=>[7,2,3,1,4,5,6]=>[3,7,2,1,4,5,6]=>[[[.,[.,.]],.],[.,[.,[.,.]]]]
[[[[[.,.],[.,.]],.],.],[.,.]]=>[7,3,1,2,4,5,6]=>[3,1,7,2,4,5,6]=>[[.,.],[[.,.],[.,[.,[.,.]]]]]
[[[[[.,[.,.]],.],.],.],[.,.]]=>[7,2,1,3,4,5,6]=>[2,7,1,3,4,5,6]=>[[.,[.,.]],[.,[.,[.,[.,.]]]]]
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[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]=>[5,4,6,3,7,2,8,1]=>[8,7,6,5,4,3,2,1]=>[[[[[[[[.,.],.],.],.],.],.],.],.]
[[[[[[[[.,.],.],.],.],.],.],.],.]=>[1,2,3,4,5,6,7,8]=>[1,2,3,4,5,6,7,8]=>[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
Map
to 132-avoiding permutation
Description
Return a 132-avoiding permutation corresponding to a binary tree.
The linear extensions of a binary tree form an interval of the weak order called the Sylvester class of the tree. This permutation is the maximal element of the Sylvester class.
The linear extensions of a binary tree form an interval of the weak order called the Sylvester class of the tree. This permutation is the maximal element of the Sylvester class.
Map
descent views to invisible inversion bottoms
Description
Return a permutation whose multiset of invisible inversion bottoms is the multiset of descent views of the given permutation.
An invisible inversion of a permutation $\sigma$ is a pair $i < j$ such that $i < \sigma(j) < \sigma(i)$. The element $\sigma(j)$ is then an invisible inversion bottom.
A descent view in a permutation $\pi$ is an element $\pi(j)$ such that $\pi(i+1) < \pi(j) < \pi(i)$, and additionally the smallest element in the decreasing run containing $\pi(i)$ is smaller than the smallest element in the decreasing run containing $\pi(j)$.
This map is a bijection $\chi:\mathfrak S_n \to \mathfrak S_n$, such that
An invisible inversion of a permutation $\sigma$ is a pair $i < j$ such that $i < \sigma(j) < \sigma(i)$. The element $\sigma(j)$ is then an invisible inversion bottom.
A descent view in a permutation $\pi$ is an element $\pi(j)$ such that $\pi(i+1) < \pi(j) < \pi(i)$, and additionally the smallest element in the decreasing run containing $\pi(i)$ is smaller than the smallest element in the decreasing run containing $\pi(j)$.
This map is a bijection $\chi:\mathfrak S_n \to \mathfrak S_n$, such that
- the multiset of descent views in $\pi$ is the multiset of invisible inversion bottoms in $\chi(\pi)$,
- the set of left-to-right maximima of $\pi$ is the set of maximal elements in the cycles of $\chi(\pi)$,
- the set of global ascent of $\pi$ is the set of global ascent of $\chi(\pi)$,
- the set of maximal elements in the decreasing runs of $\pi$ is the set of deficiency positions of $\chi(\pi)$, and
- the set of minimal elements in the decreasing runs of $\pi$ is the set of deficiency values of $\chi(\pi)$.
Map
to increasing tree
Description
Sends a permutation to its associated increasing tree.
This tree is recursively obtained by sending the unique permutation of length $0$ to the empty tree, and sending a permutation $\sigma$ of length $n \geq 1$ to a root node with two subtrees $L$ and $R$ by splitting $\sigma$ at the index $\sigma^{-1}(1)$, normalizing both sides again to permutations and sending the permutations on the left and on the right of $\sigma^{-1}(1)$ to the trees $L$ and $R$, respectively.
This tree is recursively obtained by sending the unique permutation of length $0$ to the empty tree, and sending a permutation $\sigma$ of length $n \geq 1$ to a root node with two subtrees $L$ and $R$ by splitting $\sigma$ at the index $\sigma^{-1}(1)$, normalizing both sides again to permutations and sending the permutations on the left and on the right of $\sigma^{-1}(1)$ to the trees $L$ and $R$, respectively.
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